Gap Method
Place the unrestricted items first, then put the restricted ones in the gaps.
What is it?
To keep certain items apart, arrange the others and choose distinct gaps between (and around) them for the restricted items.
Why is it useful?
"No two adjacent" constraints are awkward to count directly but immediate with gaps.
When should you think of it?
Think of it when some items must not be next to each other.
Think of this method when…
- "No two adjacent", "separated by at least one"
- Two classes of items in a row
- Binary strings without consecutive 1s
- Seating with forbidden neighbors
Core idea
items in a row create gaps; each restricted item takes its own gap.
Step-by-step playbook
- 1Arrange the unrestricted items (count their orderings).
- 2Count the gaps, including both ends.
- 3Choose gaps for the restricted items and order them if distinct.
- 4Multiply.
Worked example
Worked example
5 men and 3 women sit in a row at random. What is the probability no two women are adjacent?
- 1
Arrange the men: . That leaves gaps.
- 2
Place the 3 distinct women in distinct gaps: .
- 3
Divide by all orderings.
- Answer:
Common mistakes
- Forgetting the two end gaps.
- Not ordering distinct restricted items.
- Using linear gaps for a circle (a circle of has gaps).
Would you use this method?
How many ways can 4 people sit together as a block in a row of 8?
Practice this method
15 QuantSolve questions are tagged with this method.
Practice this method